Dynamics of a Ivlev-type predator–prey system with constant rate harvesting

Abstract

In this paper, by using the analysis of qualitative method and bifurcation theory, we investigate the dynamical properties of the Ivlev-type predator–prey model with nonzero constant prey harvesting and with or without time delay, respectively. It is shown that the system we considered can exhibit the subcritical and supercritical Hopf bifurcation. We also study the effect of the time delay on the dynamics of the system. By choosing the delay τ as a bifurcation parameter, we show that Hopf bifurcation can occur as the delay τ crosses some critical values. The direction and stability of the Hopf bifurcation are investigated by following the procedure of deriving normal form given by Faria and Magalhães. Finally, numerical simulations are performed to illustrate the obtained results.

Introduction

Mathematical models have played an important role throughout the history of ecology. Models in ecology serve a variety of purposes, which range from illustrating an idea to parameterizing a complex real-world situation. They are used to make general predictions, to guide management practices, and to provide a basis for the development of statistical tools and testable hypotheses [19], [18].

A fundamental goal of theoretical ecology is to understand how the interactions of individual organisms with each other and with the environment determine the distribution of populations and the structure of communities [3]. As we know, our ecological environment is a huge and highly complex system. This complexity arises in part from the diversity of biological species, and also from the complexity of every individual organism [11]. The dynamic behavior of predator–prey model has long been and will continue to be one of the dominant themes in both ecology and mathematical ecology due to its universal existence and importance [2], [14].

A classical predator–prey system can be written as the form [1]:

$$\dot{x}=\mathit{xf}(x)-\mathit{yg}(x\text{,}y)\text{,}\dot{y}=h[g(x\text{,}y)\text{,}v]y\text{.}$$

Where x and y are prey and predator density, respectively, f(x) the prey growth rate, g(x, y) the functional response, the prey consumption rate by an average single predator, which obviously increases with the prey consumption rate and can be influenced by the predator density, h[g(x, y), y] the per capita growth rate of predators (also known as the “predator numerical response”). The most widely accepted assumption for the numerical response is the linear one [1]:

$$h[g(x\text{,}y)\text{,}y]=\epsilon g(x\text{,}y)-\beta$$

where β is the per capita predator death rate and ε the conversion efficiency of food into offsprings.

In population dynamics, a functional response g(x, y) of the predator to the prey density refers to the change in the density of prey attached per unit time per predator as the prey density changes [21]. There is an important prey-dependent functional response – Ivlev-type, originally due to Ivlev [10]:

$$g(x\text{,}y)=1-{\mathrm{e}}^{-\gamma x}\text{.}$$

and the corresponding Ivlev-type predator–prey model takes the form:

$$\dot{x}=x(1-x)-y(1-{\mathrm{e}}^{-\gamma x})\text{,}\dot{y}=\epsilon y(1-{\mathrm{e}}^{-\gamma x})-\beta y\text{.}$$

where x and y represent population density of prey and predator at time t, respectively, ε, β, γ are positive constants, ε the conversion rate of prey captured by predator, β the deathrate of predator, and γ the efficiency of predator capture of prey.

In population dynamics, both ecologists and mathematicians are interested in model (3) [24], [13], [26], [15], [27], [30]. Existence and uniqueness of limit cycle of the Ivlev’s response predator–prey system were studied by for example Kooij and Zegling [13] and Sugie [24]. Liu et al. [15] and Xiang and Song [27] investigated the conditions for the permanence of the system and the existence and stability of a positive periodic solution. The dynamical complexity of the model with impulsive effect was considered by Wang and Wang [26]. Xiao [30] studied the global analysis of the Ivlev’s response predator–prey system.

From the point of view of human needs, the exploitation of biological resources and the harvest of population are commonly practiced in fishery, forestry and wildlife management. Concerning the conservation for the long-term benefits of humanity, there is a wide-range of interest in the use of bioeconomic modeling to gain insight in the scientific management of renewable resources like fisheries and forestries [28], [29]. At the same time, harvesting has a strong impact on the dynamic evolution of a population. Depending on the nature of applied harvesting strategy, the long-run stationary density of the population may be significantly smaller than the long-run stationary density of a population in the absence of harvesting. In the absence of harvesting, a population can be free of extinction risk; however, harvesting can lead to the incorporation of a positive extinction probability and therefore, to potential extinction in finite time. If a population is subject to a positive extinction rate then harvesting can drive the population density to a dangerously low level at which extinction becomes sure no matter how the harvester affects the population afterwards [12].

On the other hand, since the pioneering work of Volterra, time delays were already incorporated into the mathematical models of population dynamics. For a long time, it has been recognized that delays can have very complicated impact on the dynamics of a system. For example, delays can cause the loss of stability and can induce various oscillations and periodic solutions. It is well known that periodic solutions can arise through the Hopf bifurcation in delay differential equations. And a great deal of research has been devoted to the local Hopf bifurcation existence and the direction and stability of the Hopf bifurcation for a predator–prey system in delayed scalar equations. For further details, refer to Refs. [8], [22], [34], [33], [23], [6], [20], [25], [16], [17], [4], [32], [31].

In this paper, we assume that the predator in model (3) is not of commercial importance. The prey is continuously being harvested at a constant rate by a harvesting agency. The harvesting activity does not affect the predator population directly. It is obvious that the harvesting activity does reduce the predator population indirectly by reducing the availability of the prey to the predator.

Considering the nonzero prey harvesting h, we formulate the above problem as follows:

$$\dot{x}=x(1-x)-y(1-{\mathrm{e}}^{-\gamma x})-h\triangleq f_1(x\text{,}y)\text{,}\dot{y}=\beta y(\alpha -1-\alpha {\mathrm{e}}^{-\gamma x})\triangleq f_2(x\text{,}y)\text{.}$$

Where α = ε/β, h > 0 represents the rate of harvesting or removal. The corresponding delayed model with harvesting of prey at a constant rate takes the form:

$$\dot{x}(t)=x(t)(1-x(t-\tau ))-y(t)(1-{\mathrm{e}}^{-\gamma x(t)})-h\text{,}\dot{y}(t)=\beta y(t)(\alpha -1-\alpha {\mathrm{e}}^{-\gamma x(t)})\text{.}$$

The organization of this paper is as follows. In next section, we study the existence of the equilibria and various types of dynamical behavior in the small neighborhood of the equilibrium for model (4). In Section 3, we show that model (4) exhibit the subcritical and supercritical Hopf bifurcation. Based on these results, in Section 4, we consider the stability and bifurcation of the delayed Ivlev-type predator–prey model (5). In Section 5, we consider the stability and direction of the Hopf bifurcation of model (5). Finally, we give a brief discussion.